Updated on 2021/08/31

写真b

 
KAKEI Saburou
 
*Items subject to periodic update by Rikkyo University (The rest are reprinted from information registered on researchmap.)
Affiliation*
College of Science Department of Mathematics
Graduate School of Science Field of Study: Mathematics
Graduate School of Science Field of Study: Mathematics
Title*
Professor
Degree
博士(工学) ( 東京大学 )
Research Theme*
  • 主要研究テーマである「可積分系」とは、物理をはじめとする諸分野に現れる「良い」モデルの背後にある数学的構造を研究するものである。また、非線型現象全般にも興味をもっていて、2つのテーマの接点であるソリトン方程式を中心に研究を進めている。

  • Research Interests
  • Special functions

  • Nonlinear phenomena

  • Integrable systems

  • Solitons

  • Campus Career*
    • 4 2010 - Present 
      College of Science   Department of Mathematics   Professor
    • 4 2010 - Present 
      Graduate School of Science   Field of Study: Mathematics   Professor
    • 4 2010 - Present 
      Graduate School of Science   Field of Study: Mathematics   Professor
    • 4 2007 - 3 2010 
      College of Science   Department of Mathematics   Associate Professor
    • 4 2003 - 3 2007 
      College of Science   Department of Mathematics   Associate Professor (as old post name)
    • 4 2001 - 3 2003 
      College of Science   Department of Mathematics   Lecturer
     

    Research Areas

    • Natural Science / Applied mathematics and statistics

    • Natural Science / Mathematical analysis  / Integrable Systems

    • Natural Science / Basic mathematics

    Research History

    • 4 2010 - Present 
      RIKKYO UNIVERSITY   Graduate School of Science Field of Study: Mathematics   Professor

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    • 4 2010 - Present 
      RIKKYO UNIVERSITY   Graduate School of Science Field of Study: Mathematics   Professor

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    • 4 2010 - Present 
      RIKKYO UNIVERSITY   College of Science Department of Mathematics   Professor

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    • 4 2007 - 3 2010 
      RIKKYO UNIVERSITY   College of Science Department of Mathematics   Associate Professor

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    • 4 2003 - 3 2007 
      RIKKYO UNIVERSITY   College of Science Department of Mathematics   Associate Professor (as old post name)

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    • 4 2001 - 3 2003 
      RIKKYO UNIVERSITY   College of Science Department of Mathematics   Lecturer

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    • 4 1999 - 3 2001 
      Waseda University   School of Science and Engineering

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    • 4 1998 - 3 1999 
      日本学術振興会   日本学術振興会特別研究員

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    • 4 1995 - 3 1998 
      The University of Tokyo   Graduate School of Mathematical Sciences

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    Education

    • 4 1992 - 3 1995 
      The University of Tokyo   The Graduate School of Engineering   Department of Applied Physics

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      Country: Japan

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    • 4 1990 - 3 1992 
      The University of Tokyo   The Graduate School of Engineering   Department of Applied Physics

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      Country: Japan

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    • 4 1988 - 3 1990 
      The University of Tokyo   Faculty of Engineering   Department of Applied Physics

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      Country: Japan

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    Committee Memberships

    • 9 2006 - Present 
      日本物理学会   JPSJ編集委員

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      Committee type:Academic society

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    • 4 2006 - 3 2007 
      日本応用数理学会   評議員

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      Committee type:Academic society

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    • 4 2004 - 3 2005 
      日本数学会   代議員

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      Committee type:Academic society

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    • 11 2002 - 10 2003 
      日本物理学会   領域11世話人

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      Committee type:Academic society

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    Papers

    • Toda lattice hierarchy and soliton equations on square lattice Peer-reviewed

      Saburo Kakei

      Journal of Physics A: Mathematical and Theoretical54 ( 7 ) 074001 - 074001   19 2 2021

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      Language:English   Publishing type:Research paper (scientific journal)   Publisher:IOP Publishing  

      DOI: 10.1088/1751-8121/ab88e2

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      Other Link: https://iopscience.iop.org/article/10.1088/1751-8121/ab88e2/pdf

    • Linearization of the box-ball system: an elementary approach Peer-reviewed

      S. Kakei, J.J.C. Nimmo, S. Tsujimoto, R. Willox

      Journal of Integrable Systems3   1 - 32   2018

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    • Toda lattice hierarchy and Goldstein-Petrich flows for plane curves Peer-reviewed

      Kenji Kajiwara, Saburo Kakei

      Comment. Math. Univ. St. Pauli64   29 - 45   5 2015

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      Language:English   Publishing type:Research paper (scientific journal)  

      A relation between the Goldstein-Petrich hierarchy for plane curves and the Toda lattice hierarchy is investigated. A representation formula for plane curves is given in terms of a special class of τ-functions of the Toda lattice hierarchy. A representation formula for discretized plane curves is also discussed.

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    • Yang-Baxter Maps from the Discrete BKP Equation Peer-reviewed

      Saburo Kakei, Jonathan J. C. Nimmo, Ralph Willox

      SYMMETRY INTEGRABILITY AND GEOMETRY-METHODS AND APPLICATIONS6   028   2010

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      Language:English   Publishing type:Research paper (scientific journal)   Publisher:NATL ACAD SCI UKRAINE, INST MATH  

      We construct rational and piecewise-linear Yang-Baxter maps for a general N-reduction of the discrete BKP equation.

      DOI: 10.3842/SIGMA.2010.028

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    • YANG-BAXTER MAPS AND THE DISCRETE KP HIERARCHY Peer-reviewed

      S. Kakei, J. J. C. Nimmo, R. Willox

      GLASGOW MATHEMATICAL JOURNAL51A ( A ) 107 - 119   2 2009

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      Language:English   Publishing type:Research paper (scientific journal)   Publisher:CAMBRIDGE UNIV PRESS  

      We present a systematic construction of the discrete KP hierarchy in terms of Sato-Wilson-type shift operators. Reductions of the equations in this hierarchy to 1+1-dimensional integrable lattice systems are considered, and the problems that arise with regard to the symmetry algebra underlying the reduced systems as well as the Ultradiscretizability of these systems are discussed. A scheme for constructing ultradiscretizable reductions that give rise to Yang-Baxter maps is explained in two explicit examples.

      DOI: 10.1017/S0017089508004825

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    • The Rational qKZ Equation and Shifted Non-Symmetric Jack Polynomials Peer-reviewed

      Saburo Kakei, Michitomo Nishizawa, Yoshihisa Saito, Yoshihiro Takeyama

      SYMMETRY INTEGRABILITY AND GEOMETRY-METHODS AND APPLICATIONS5   010   2009

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      Language:English   Publishing type:Research paper (scientific journal)   Publisher:NATL ACAD SCI UKRAINE, INST MATH  

      We construct special solutions to the rational quantum Knizhnik-Zamolodchikov equation associated with the Lie algebra gl(N). The main ingredient is a special class of the shifted non-symmetric Jack polynomials. It may be regarded as a shifted version of the singular polynomials studied by Dunkl. We prove that our solutions contain those obtained as a scaling limit of matrix elements of the vertex operators of level one.

      DOI: 10.3842/SIGMA.2009.010

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    • Difference equations with an invariant expressed in terms of the hypergeometric function Peer-reviewed

      Takeshi Sakaki, Saburo Kakei

        17 ( 4 ) 455 - 462   12 2007

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    • Discrete stochastic processes and q-hypergeometric functions (jointly worked) Peer-reviewed

      Takeshi Ga, Saburo Kakei, Yasushi Kitane

        17 ( 4 ) 463 - 468   12 2007

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    • The sixth Painleve equation as similarity reduction of (gl)over-cap(3) generalized Drinfel'd-Sokolov hierarchy Peer-reviewed

      Saburo Kakei, Tetsuya Kikuchi

      LETTERS IN MATHEMATICAL PHYSICS79 ( 3 ) 221 - 234   3 2007

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      Language:English   Publishing type:Research paper (scientific journal)   Publisher:SPRINGER  

      Scaling symmetry of (gl) over cap (n)-type Drinfel'd-Sokolov hierarchy is investigated. Applying similarity reduction to the hierarchy, one can obtain the Schlesinger equation with (n + 1) regular singularities. Especially in the case of n = 3, the hierarchy contains the three-wave resonant system and the similarity reduction gives the generic case of the Painleve VI equation. We also discuss Weyl group symmetry of the hierarchy.

      DOI: 10.1007/s11005-007-0144-4

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    • A q-analogue of gl<SUB>3</SUB> hierarchy and q-Painlevé VI (共著) Peer-reviewed

      Saburo Kakei, Tetsuya Kikuchi

      Journal of Physics A: Mathematical and General39 ( 39 ) 12179 - 12190   29 9 2006

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      DOI: 10.1088/0305-4470/39/39/S11

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    • Toroidal Lie algebra and bilinear identity of the self-dual Yang-Mills hierarchy Peer-reviewed

      Saburo Kakei

      Bilinear Integrable Systems: From Classical to Quatum, Continuous to Discrete201   137 - 146   2006

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      Language:English   Publishing type:Research paper (international conference proceedings)   Publisher:SPRINGER  

      Bilinear identity associated with the self-dual Yang-Mills hierarchy is discussed by using a fermionic representation of the toroidal Lie algebra. sl(2)(tor)

      DOI: 10.1007/978-1-4020-3503-6_13

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    • SOLUTIONS OF A DERIVATIVE NONLINEAR SCHRODINGER HIERARCHY AND ITS SIMILARITY REDUCTION Peer-reviewed

      Saburo Kakei, Tetsuya Kikuchi

      GLASGOW MATHEMATICAL JOURNAL47A   99 - 107   6 2005

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      Language:English   Publishing type:Research paper (scientific journal)   Publisher:CAMBRIDGE UNIV PRESS  

      The hierarchy structure of a derivative nonlinear Schrodinger equation is investigated in terms of the Sato-Segal-Wilson formulation. Special solutions are constructed as ratios of Wronski determinants. Relations to the Painleve IV and the discrete Painleve I are discussed by applying a similarity reduction.

      DOI: 10.1017/S0017089505002326

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    • Affine lie group approach to a derivative nonlinear Schrodinger equation and its similarity reduction Peer-reviewed

      S Kakei, T Kikuchi

      INTERNATIONAL MATHEMATICS RESEARCH NOTICES2004 ( 78 ) 4181 - 4209   2004

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      Language:English   Publishing type:Research paper (scientific journal)   Publisher:OXFORD UNIV PRESS  

      DOI: 10.1155/S1073792804141573

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    • Similarity reduction of the modified Yajima-Oikawa equation Peer-reviewed

      Tetsuya Kikuchi, Takeshi Ikeda, Saburo Kakei

      Journal of Physics A: Mathematical and General36 ( 45 ) 11465 - 11480   14 11 2003

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      We study a similarity reduction of the modified Yajima-Oikawa hierarchy. The hierarchy is associated with a non-standard Heisenberg subalgebra in the affine Lie algebra of type A2(1)The system of equations for self-similar solutions is presented as a Hamiltonian system of degree of freedom 2, and admits a group of Bäcklund transformations isomorphic to the affine Weyl group of type A2(1). We show that the system is equivalent to a two-parameter family of the fifth Painlevé equation.

      DOI: 10.1088/0305-4470/36/45/008

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    • Hierarchy of (2+1)-dimensional nonlinear Schrodinger equation, self-dual Yang-Mills equation, and toroidal Lie algebras Peer-reviewed

      S Kakei, T Ikeda, K Takasaki

      ANNALES HENRI POINCARE3 ( 5 ) 817 - 845   2002

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      Language:English   Publishing type:Research paper (scientific journal)   Publisher:BIRKHAUSER VERLAG AG  

      The hierarchy structure associated with a (2 + 1)-dimensional Nonlinear Schrodinger equation is discussed as an extension of the theory of the KP hierarchy. Several methods to construct special solutions are given. The relation between the hierarchy and a representation of toroidal Lie algebras are established by using the language of free fermions. A relation to the self-dual Yang-Mills equation is also discussed.

      DOI: 10.1007/s00023-002-8638-1

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    • A differential-difference system related to toroidal Lie algebra Peer-reviewed

      Saburo Kakei, Yasuhiro Ohta

      Journal of Physics A: Mathematical and General34 ( 48 ) 10585 - 10592   7 12 2001

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      Language:English   Publishing type:Research paper (scientific journal)  

      We present a novel differential-difference system in (2 + 1)-dimensional space-time (one discrete, two continuum), arising from Bogoyavlensky's (2 + 1)-dimensional Korteweg-de Vries hierarchy. Our method is based on the bilinear identity of the hierarchy, which is related to the vertex-operator representation of the toroidal Lie algebra, s-fraktur sign and l-fractur signtor2.

      DOI: 10.1088/0305-4470/34/48/322

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    • Algegraic structure of coupled KP hierarchy

      Saburo Kakei

        ( 11ME-S4 ) 178 - 182   3 2000

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      Language:Japanese   Publishing type:Research paper (bulletin of university, research institution)  

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    • Algebraic aspects of quantum Calogero models (共著)

      Saburo Kakei, Yusuke Kato

      Special Functions: Proceeding of the International Workshop   125 - 139   15 1 2000

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    • Dressing method and the coupled KP hierarchy Peer-reviewed

      S Kakei

      PHYSICS LETTERS A264 ( 6 ) 449 - 458   1 2000

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      Language:English   Publishing type:Research paper (scientific journal)   Publisher:ELSEVIER SCIENCE BV  

      The coupled KP hierarchy, introduced by Hirota and Ohta, is investigated by using the dressing method. It is shown that the coupled KP hierarchy can be reformulated as a reduced case of the 2-component KP hierarchy. (C) 2000 Elsevier Science B.V. All rights reserved.

      DOI: 10.1016/S0375-9601(99)00848-8

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    • Experiment on shock wave in a random LC circuit Peer-reviewed

      H Asano, S Kakei, S Ishiwata, S Watanabe

      JOURNAL OF THE PHYSICAL SOCIETY OF JAPAN68 ( 10 ) 3208 - 3213   10 1999

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      Language:English   Publishing type:Research paper (scientific journal)   Publisher:PHYSICAL SOCIETY JAPAN  

      Shock wave in a random system has been investigated theoretically and experimentally. A nonlinear LC ladder circuit with random inhomogeneity of linear inductor in series branch has been studied. The circuit is equivalent to a random Toda lattice. It is shown theoretically that wave propagation in the system is described by the Burgers equation in the case of strong randomness with weak nonlinearity. The transition from an oscillatory shock wave to a shock wave has been observed on a random LC ladder circuit experimentally. The velocity and width of shock wave agree with the theoretical prediction from the Burgers equation in a region of strong randomness.

      DOI: 10.1143/JPSJ.68.3208

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    • Orthogonal and symplectic matrix integrals and coupled KP hierarchy Peer-reviewed

      S Kakei

      JOURNAL OF THE PHYSICAL SOCIETY OF JAPAN68 ( 9 ) 2875 - 2877   9 1999

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      Language:English   Publishing type:Research paper (scientific journal)   Publisher:PHYSICAL SOCIETY JAPAN  

      Orthogonal and symplectic matrix integrals are investigated. It is shown that the matrix integrals can be considered as a tau-function of the coupled KP hierarchy, whose solution can be expressed in terms of pfaffians.

      DOI: 10.1143/JPSJ.68.2875

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    • Coupled KP hierarchy and matrix integrals

      Saburo Kakei

        ( 10ME-S1 ) 69 - 74   5 1999

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    • Intertwining operators for a degenerate double affine Hecke algebra and multivariable orthogonal polynomials Peer-reviewed

      S Kakei

      JOURNAL OF MATHEMATICAL PHYSICS39 ( 9 ) 4993 - 5006   9 1998

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      Language:English   Publishing type:Research paper (scientific journal)   Publisher:AMER INST PHYSICS  

      Several properties of the multivariable Hermite and Laguerre polynomials associated with the quantum Calogero models are investigated by using the operators that intertwine representations of a degenerate version of the double affine Hecke algebra. As applications, raising operators and shift operators for the polynomials are constructed in a unified manner. (C) 1998 American Institute of Physics.

      DOI: 10.1063/1.532505

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    • Algegraic structure of quantum Calogero models

      Saburo Kakei

        ( 9ME-S2 ) 29 - 34   5 1998

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    • Quantum Calogero models and the degenerate double affine Hecke algebra

      筧 三郎

      京都大学 数理解析研究所 講究録1036   65 - 87   4 1998

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    • Reply to the remark by W. Malfliet on "Shock wave in a nonlinear LC circuit" Peer-reviewed

      S Watanabe, S Ishiwata, J Ehara, S Kakei

      JOURNAL OF THE PHYSICAL SOCIETY OF JAPAN67 ( 2 ) 702 - 702   2 1998

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      Language:English   Publishing type:Research paper (scientific journal)   Publisher:PHYSICAL SOC JAPAN  

      DOI: 10.1143/JPSJ.67.702

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    • An orthogonal basis for the B<SUB>N</SUB>-type Calogero model Peer-reviewed

      Saburo Kakei

      Journal of Physics A: Mathematical and General30 ( 15 ) L535 - L541   7 8 1997

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      DOI: 10.1088/0305-4470/30/15/009

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    • On long-range interaction models of 1/r<SUP>2</SUP>-type

      Saburo Kakei

        1005   72 - 94   6 1997

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    • 量子Calogero模型

      数理科学35 ( 3 ) 54   4 1997

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      Language:Japanese   Publishing type:Research paper (other academic)  

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    • Quantum Calogero model

      Mathematial Sciences35 ( 3 ) 54   4 1997

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    • Common algebraic structure for the Calogero-Sutherland models Peer-reviewed

      Saburo Kakei

      Jornal of Physics A: Mathematical and General29 ( 24 ) L619 - L624   21 12 1996

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      DOI: 10.1088/0305-4470/29/24/002

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    • Toda lattice hierarchy and Zamolodchikov's conjecture Peer-reviewed

      S Kakei

      JOURNAL OF THE PHYSICAL SOCIETY OF JAPAN65 ( 2 ) 337 - 339   2 1996

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      Language:English   Publishing type:Research paper (scientific journal)   Publisher:PHYSICAL SOC JAPAN  

      In this letter, we show that the Fredholm determinant D(lambda;t), introduced by Zamolodchikov in his study of 2D polymers, is the continuum limit of the soliton solution for the Toda lattice hierarchy with a 2-periodic reduction condition.

      DOI: 10.1143/JPSJ.65.337

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    • BILINEARIZATION OF A GENERALIZED DERIVATIVE NONLINEAR SCHRODINGER-EQUATION Peer-reviewed

      S KAKEI, N SASA, J SATSUMA

      JOURNAL OF THE PHYSICAL SOCIETY OF JAPAN64 ( 5 ) 1519 - 1523   5 1995

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      Language:English   Publishing type:Research paper (scientific journal)   Publisher:PHYSICAL SOC JAPAN  

      A generalized derivative nonlinear Schrodinger equation, iq(t)+q(xx)+2i gamma\q\(2)q(x)+2i(gamma-1)q(2)q(x)(*)+(gamma-1)(gamma-2)\q\(4)q=0, is studied by means of Hirota's bilinear formalism. Soliton solutions are constructed as quotients of Wronski-type determinants. A relationship between the bilinear structure and gauge transformation is also discussed.

      DOI: 10.1143/JPSJ.64.1519

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    • Bilinear structure of nonlinear Schrodinger-type equations (jointly worked)

      Saburo Kakei, Narimasa Sasa, Junkichi Satsuma

        889   101 - 112   11 1994

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    • A generalization of Grassmann hierarchy (jointly worked)

      Saburo Kakei, Junkichi Satsuma

        868   85 - 97   4 1994

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    • MULTISOLITON SOLUTIONS OF A COUPLED SYSTEM OF THE NONLINEAR SCHRODINGER-EQUATION AND THE MAXWELL-BLOCH EQUATIONS Peer-reviewed

      S KAKEI, J SATSUMA

      JOURNAL OF THE PHYSICAL SOCIETY OF JAPAN63 ( 3 ) 885 - 894   3 1994

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      Language:English   Publishing type:Research paper (scientific journal)   Publisher:PHYSICAL SOC JAPAN  

      Multi-soliton solutions of a coupled system of the nonlinear Schrodinger equation and the Maxwell-Bloch equations are given. Solutions of the system are explicitly constructed as a quotient of Casorati-type determinants. By using explicit form of the soliton solutions, the influence of the MB-term on the speed of soliton is evaluated.

      DOI: 10.1143/JPSJ.63.885

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    Misc.

    • 線形代数の持つ力 Invited

      筧三郎

      数学セミナー50 ( 4 ) 12 - 17   3 2021

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      Language:Japanese   Publishing type:Article, review, commentary, editorial, etc. (trade magazine, newspaper, online media)   Publisher:日本評論社  

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    • “1/6公式” から超幾何関数へ

      筧三郎

      数学セミナー56 ( 4 ) 36   3 2017

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      Language:Japanese   Publishing type:Article, review, commentary, editorial, etc. (trade magazine, newspaper, online media)   Publisher:日本評論社  

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    • 連続性

      筧三郎

      数学セミナー46 ( 6 ) 26 - 30   1 6 2007

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      Language:Japanese   Publishing type:Article, review, commentary, editorial, etc. (trade magazine, newspaper, online media)   Publisher:日本評論社  

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    • Random matrices and integrable systems

      Saburo Kakei

      Mathematical Sciences ( 524 ) 24 - 29   1 2 2007

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      Language:Japanese   Publishing type:Article, review, commentary, editorial, etc. (trade magazine, newspaper, online media)   Publisher:Saiensu sha Co. Ltd. Publishers  

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    • ベクトルとベクトル空間

      筧三郎

      数学セミナー45 ( 4 ) 27 - 31   1 4 2006

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    • 運動方程式からの微分積分

      筧三郎

      数学セミナー44 ( 4 ) 26 - 29   1 4 2005

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      Language:Japanese   Publishing type:Article, review, commentary, editorial, etc. (trade magazine, newspaper, online media)   Publisher:日本評論社  

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    • 可積分系という「おもちゃ」

      筧三郎

      数学セミナー42 ( 4 ) 47 - 49   1 4 2003

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      Language:Japanese   Publishing type:Article, review, commentary, editorial, etc. (trade magazine, newspaper, online media)   Publisher:日本評論社  

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    • Quantum Calogero models

      Saburo Kakei

      Mathematical Sciences ( 405 ) 54 - 59   1 3 1997

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      Language:Japanese   Publishing type:Article, review, commentary, editorial, etc. (trade magazine, newspaper, online media)   Publisher:Saiensu sha Co. Ltd. Publishers  

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    Books and Other Publications

    • Vector Analysis

      Saburo Kakei, Gen Yoneda( Role: Joint author)

      8 2018  ( ISBN:4781914268

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      Language:Japanese

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    • Handbook of Self-Organization

      ( Role: Other)

      11 2009  ( ISBN:9784860432645

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      Language:Japanese Book type:Dictionary, encyclopedia

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    • 理工系のための 解く! 線形代数

      筧三郎, 西成活裕( Role: Sole author)

      講談社サイエンティフィク  16 11 2007  ( ISBN:406155767X

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      Language:Japanese Book type:Textbook, survey, introduction

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    • 応用数学ハンドブック

      藤原毅夫, 平尾公彦, 久田俊明, 広瀬啓吉( Role: Other ,  線形代数)

      丸善  3 2005  ( ISBN:4621075292

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      Language:Japanese Book type:Dictionary, encyclopedia

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    • 工科系 線形代数

      筧 三郎( Role: Sole author)

      数理工学社  10 2002  ( ISBN:4901683020

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      Language:Japanese Book type:Textbook, survey, introduction

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    Research Projects

    • 諸分野に現れるタウ関数の構造

      日本学術振興会  科学研究費助成事業 基盤研究(C) 

      筧 三郎

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      4 2016 - 3 2021

      Grant number:16K05184

      Grant amount:\4680000 ( Direct Cost: \3600000 、 Indirect Cost:\1080000 )

      本研究では,可積分系の研究における「タウ関数」という視点を広げるとともに,その概念と諸分野との関連を考察することで,可積分系の世界をさらに広げていくことを目標としている。2019年度は,離散可積分系と関連するテーマを研究し,昨年度からの研究を発展する形での成果が得られた。特に,格子KdV方程式,格子ブシネスク方程式とKP階層のタウ関数との関係について,昨年度では証明が完成していなかった部分をまとめることができた。具体的には,オランダのグループによって研究されてきたタイプの格子KdV方程式,格子ブシネスク方程式が,以前の梶原(九大IMI)との共同研究で得られた離散戸田階層の定式化から得られることを示すことができ,それに伴い,戸田階層のタウ関数との関係も確立することができた。また,戸田階層の代数関数解について,高崎の先行結果を利用することで,格子KdV方程式,格子ブシネスク方程式の解を構成することができた。この場合,よく知られているように,タウ関数はリーマン・テータ関数を用いて表される。上記の結果を証明する際には膨大な代数的計算が必要となるが,その部分には,計算機代数システム Risa/Asir を利用した。上記の結果については論文を投稿中である。2019年度に参加した研究集会でも,これらの結果を紹介した。
      また,昨年度からの引き続きのテーマである「Jeu de taquin と離散戸田格子」,「交通流モデルとGKZ超幾何系」という話題については,進展はあったものの,最終的な解決までには至っていない。本年度も引き続き研究を続けていこうと考えている。

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    • Extension of special functions from the viewpoint of integrable systems

      Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C) 

      KAKEI Saburo, SAITO Yoshihisa, WILLOX Ralph

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      4 2011 - 3 2016

      Grant number:23540252

      Grant amount:\4810000 ( Direct Cost: \3700000 、 Indirect Cost:\1110000 )

      This research project aims to extend the theory of integrable systems and special functions. We have focused mainly on applications to differential geometry and combinatorics, and obtained the following results;
      (1) Goldstein and Petrich introduced a hierarchy of equations that governs isoperimetric deformation of plane curves. We have revealed that the Goldstein-Petrich hierarchy is indeed a reduced case of the Toda lattice hierarchy. (2) We have studied rigged configurations from the viewpoint of soliton cellular automaton. We have obtained alternative proof of the linearization property of the time-evolution, which is simpler than the known proofs.
      We have obtained still other preliminary results that are related to applications of integrable systems to differential geometry and combinatorics. We will investigate them further in near future.

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    • 古典可積分系と量子可積分系の接点の探求

      日本学術振興会  科学研究費助成事業 

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      4 2007 - 3 2010

      Grant type:Competitive

      可積分系の研究は,大別すれば「古典可積分系」,「量子可積分系」の2つに分類できる。両者の間には単に「正準量子化」でつながること以上の結びつきがあり,より深い意味での「量子・古典対応」が存在していると思われる。そこで本研究では,この「量子・古典対応」の背後にある数学的構造を解明することを目的とする。

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    • 可積分系における楕円ワイル群対称性

      日本学術振興会  科学研究費助成事業 

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      4 2004 - 3 2007

      Grant type:Competitive

      これまでの可積分系研究の中で見出されてきたワイル群の構造を,「楕円ワイル群」というキーワードの下に拡張し,新たな可能性を切り開くことを目的とする。

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    • 高次元における非線形可積分系の構造

      日本学術振興会  科学研究費助成事業 

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      4 2002 - 3 2004

      Grant type:Competitive

      従来のソリトン理論では,KdV方程式,非線形シュレディンガー方程式のような(1+1)次元の時空における偏微分方程式,KP方程式,Davey-Stewartson方程式のような(2+1)次元の時空における偏微分方程式が研究されてきた。本研究では,特に「トロイダル・リー代数」の対称性に注目し,高次元における非線形可積分系の構造を解明することを目的とする。

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    • パフィアン型ソリトン方程式の構造と諸分野への応用

      日本学術振興会  科学研究費助成事業 

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      4 2000 - 3 2002

      Grant type:Competitive

      KdV方程式,KP方程式に代表されるソリトン方程式の理論においては,特殊解が行列式で表わされることが本質的であった。そこで本研究では,パフィアンで表わされる特殊解をもつソリトン方程式に対して,その構造と諸分野への応用を探ることを目的とする。

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    • 量子可積分系と多変数直交多項式

      早稲田大学  受託研究(一般受託研究) 

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      4 1999 - 3 2000

      Grant type:Competitive

      特定課題研究

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    • 量子可積分系の代数的構造の究明と物理的情報の抽出

      日本学術振興会  科学研究費助成事業 特別研究員奨励費 

      筧 三郎

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      1998 - 1998

      Grant number:98J03759

      Grant amount:\1200000 ( Direct Cost: \1200000 )

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    • Theory of Integrable Systems and Its Application

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      Grant type:Competitive

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    • 可積分系の理論とその応用

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      Grant type:Competitive

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